arXiv:1412.3685v2 [math.CO] 21 May 2018 ebgnb iigafwdfiiin ewl s hogot An throughout. use will we definitions few a giving orientation by begin We Introduction 1 dei uhawyta eoti odrce yls hsotiiga obtaining thus cycles, directed no obtain we that way a such in edge ∗ PR rn EP/P504872/1 Grant EPSRC ietv dnicto faylcoinain n lones and orientations acyclic of identification bijective cci rettoso opeebpriegah ro c a removed. of or or added edge graph, one bipartite with a bipartite of orientations acyclic xaddi nte paper. another in expanded seult h oyBrolinumber poly-Bernoulli the to equal is e hs ouincnb dnie ihtepoly-Bernoulli the the with index, identified negative counti be with a can described solution Brewbaker whose Chet lem 2008, Berunoulli In as logarithms. polylogarithms to to do relation same the much bear nvriyo tAndrews St of University cci rettosadpoly-Bernoulli and orientations Acyclic u oml hw httenme faylcoinain of orientations acyclic of number the of that number shows formula the Our for formulae give to is paper this of aim main The emk oermrso h otx forrsl,wihare which result, our of context the on remarks some make We n19,Msnb aeodefined Kaneko Masanobu 1997, In fa nietdgraph undirected an of .J Cameron J. P. iyUiest London University City .U Schumacher U. R. oeu matrices lonesum numbers Abstract G 1 sa sineto ieto oeach to direction of assignment an is iyUiest London University City B oyBrolinumbers poly-Bernoulli n ( − 1 . n 2 ) eas ieasimple a give also We . ∗ .A Glass A. C. mmatrices. um numbers numbers gprob- ng omplete which , K n 1 ,n acyclic 2 n acyclic directed graph. Let a(G) be the number of acyclic orientations of a graph G. There is always at least one acyclic orientation of G obtained by ordering the vertices of G and orienting edges from smaller to larger vertex index. Next we define Kn1,n2 to be the complete bipartite graph on n1 + n2 vertices to be the graph whose vertices are partitioned into sets of sizes n1 and n2, having all possible edges between these two sets and none within them. We denote by S(n, k) the Stirling number of the second kind which counts the number of ways to partition a set of n objects into k non-empty subsets.

2 The number of acyclic orientations of cer- tain graphs

Our main results are given in the next three theorems. Theorem 2.1 The number of acyclic orientations of the complete bipartite graph Kn1,n2 is

min{n1+1,n2+1} 2 (k − 1)! S(n1 +1,k)S(n2 +1,k), kX=1 where S denotes Stirling numbers of the second kind.

Theorem 2.2 Let G be the graph obtained from Kn1,n2 by adding an edge e1 joining two vertices in the bipartite block of size n1, where n1 > 1. Then

a(G)= a(Kn1,n2 + e1)= a(Kn1,n2 )+ a(Kn1−1,n2 ).

Theorem 2.3 Let G be the graph obtained by deleting an edge from Kn1,n2 . Then 1 a(G)= a(K 1 2 − e)= a(K 1 2 ) − X, n ,n n ,n 2 where

min{n1,n2}+1 2 X =1+ ((k − 2)!) [(2k − 3)S(n1 +1,k)S(n2 +1,k) kX=2 −(k − 2)(S(n1 +1,k)S(n2,k)+ S(n1,k)S(n2 +1,k))

−S(n1,k)S(n2,k)]. We will prove these three theorems in the next three subsections.

2 2.1 Proof of Theorem 2.1 Let A and B be the two bipartite blocks; we will imagine their vertices as coloured amber and blue respectively. Now any acyclic orientation of the graph can be obtained by ordering the vertices and making the edges point from smaller to greater. If we do this, we will have alternating amber and blue intervals; the ordering within each interval is irrelevant in identifying the orientation, but the ordering of the intervals themselves matters. In terms of structure for a given orientation, call two points a1, a2 ∈ A equivalent if the orientations of {a1, b} and {a2, b} are the same for all b ∈ B. Points of A are equivalent if and only if they are not separated by a point of B in any ordering giving rise to the acyclic orientation. Similarly for B. This gives us the intervals, which are interleaved. It is left to count alternating intervals. To get around the problem that the first interval in the ordering might be in either A or B, and similarly for the last interval, we use the following trick. Add a dummy amber vertex a0 to A and a dummy blue vertex b0 to B. Now partition A ∪{a0} and B ∪{b0} into the same number, say k, of intervals. This can be done in S(n1 +1,k)S(n2 +1,k) ways. Now we order the intervals so that

• the interval containing a0 is first; • the colours of the intervals alternate;

• the interval containing b0 is last. This can be done in (k − 1)!2 ways. Finally, delete the dummy points. Summing over k gives the total number claimed.

2.2 Proof of Theorem 2.2

Let G be the graph consisting of Kn1,n2 (with bipartite blocks A and B) together with an edge e joining two vertices in A. According to the deletion-contraction formula [5, p.172],

a(G)= a(G − e)+ a(G/e)= a(Kn1,n2 )+ a(Kn1−1,n2 ), as required.

3 2.3 Proof of Theorem 2.3 Deleting an edge is a little more difficult. Suppose that we calculate the number X of acyclic orientations of Kn1,n2 which remain acyclic when a given edge e = {a, b} is flipped. (This number clearly does not depend on the chosen edge.) Then the number of acyclic orientations of G = Kn1,n2 − e 1 1 is a(Kn1,n2 )− 2 X. For if we call this number Y , then 2 X of the acyclic orien- tations of G extend to two acyclic orientations of Kn1,n2 , while the remaining 1 1 Y − 2 X extend to a unique acyclic orientation; so a(Kn1,n2 )= X +(Y − 2 X), giving the result. It thus remains to verify the formula for X given in the statement of the theorem. We follow the construction in the proof of Theorem 2.1. The edge e can be flipped in an orientation if and only if the part of the partition of B containing b immediately precedes or follows the part of the partition of A containing a, in the corresponding vertex ordering. (If a part of B, containing a vertex b′ say, and a part of A, containing a vertex a′, intervene, then we have arcs (a, b′), (b′, a′) and (a′, b), so the arc (a, b) is forced. Similarly in the other case.) If k = 1, then all edges are directed from A to B, and(a, b) can be flipped. So this contributes 1 to the sum. Suppose that k> 2. We distinguish four cases, according as a0 and a are or are not in the same part, and similarly for b0 and b. Of the S(n1 +1,k) partitions of A ∪{a0}, S(n1,k) have a0 and a in the same part: this is found by regarding a0 and a as the same element, partitioning the resulting set of size n1, and then separating them again.

Case 1 a0 and a in the same part, b0 and b, in the same part. Since k > 1, the parts containing a0 and b0, and hence the parts containing a and b, are not consecutive, so the contribution from this case is 0.

Case 2 a0 and a in the same part, b0 and b not. There are S(n1,k)(S(n2+ 1,k) − S(n2,k)) pairs of partitions with this property. Now the part contain- ing b must come immediately after the part containing a, so there are only (k − 2)! orderings of the parts of B, while still (k − 1)! for the parts of A.

Case 3 b0 and b in the same part, a0 and a not. This case is the same as Case 2, with n1 and n2 interchanged.

4 Case 4 a0 and a in different parts, b0 and b in different parts. There are (S(n1 + 1,k) − S(n1,k))(S(n2 + 1,k) − S(n2,k)) such pairs of parti- tions. Now the parts containing a and b must be adjacent, so must occur as (3, 2), (3, 4), (5, 4),..., or (2k − 1, 2k − 2) in the ordering of parts: there are (2k − 3) possibilities. Once one possibility has been chosen, the position of two parts for both A and B are fixed, so there are ((k − 2)!)2 possible orderings. Combining all of the above terms and rearranging, gives the value of X, completing the proof.

2.4 Some numerical values It is instructive to view the numerical values of the number of acyclic orien- tations of bipartite graphs Kn1,n2 . When n1 = 1, the graph is a , and n we have a(K1,n)=2 . For n1 between 2 and 7 Table 1 gives the number of acyclic orientations of the complete bipartite graphs and Tables 2 and 3 those graphs with an edge added or removed, calculated from the formulae in Theorems 2.1, 2.2 and 2.3. In Table 2 for Kn1,n2 + e1, the added edge e1 is in the bipartite block of size n1. All of these values have been checked by calculating the of the graph. (A theorem of Stan- ley [5] asserts that the number of acyclic orientations of an n-vertex graph n G is (−1) PG(−1), where PG is the chromatic polynomial of G.)

n1 \ n2 2 3 4 5 6 7 2 14 46 146 454 1394 4246 3 230 1066 4718 20266 85310 4 6902 41506 237686 1315666 5 329462 2441314 17234438 6 22934774 22934774 7 2193664790

Table 1: The number of acyclic orientations of Kn1,n2

n Note that as well as the formula a(K1,n)=2 we have a(K2,n + e1) = n 2 · 3 . This is because the graph K2,n + e1 consists of n triangles sharing a common edge e1, there are two ways to orient the edge e1, and then three

5 n1 \ n2 2 3 4 5 6 7 2 18 54 162 486 1458 4374 3 60 276 1212 5172 21660 89556 4 192 1296 7968 46224 257952 1400976 5 600 5784 48408 370968 2679000 18550104 6 1848 24984 279192 2770776 25376088 219463704 7 5640 105576 1553352 19675752 225164040 2395894056

Table 2: The number of acyclic orientations of Kn1,n2 + e1

n1 \ n2 2 3 4 5 6 7 2 8 28 92 292 908 2788 3 152 736 3344 14608 62192 4 5000 30952 180632 1012936 5 253352 1915672 13715144 6 18381608 164501368 7 1812141032

Table 3: The number of acyclic orientations of Kn1,n2 − e

6 ways to choose the orientations of the remaining edges of each triangle to avoid a . Putting these two results together in Theorem 2.2 gives us n n a(K2,n)=2 · 3 − 2 . Is there a closed formula for a(Kn1,n2 ) in general?

3 Poly-Bernoulli numbers and lonesum ma- trices

The formulae for the number of acyclic orientations of a bipartite graph

Kn1,n2 in Theorem 2.1 appear to be obscure. However, we now show that it is actually the poly-Bernoulli number in the variables n1 and n2, as defined by Kaneko, for which another combinatorial interpretation was found by Brewbaker. In this section we explain the connections.

3.1 Poly-Bernoulli numbers This is only a very brief introduction to the poly-Bernoulli numbers, which were introduced by Masanobu Kaneko [3] in 1997. Kaneko gave the following definitions. Let ∞ zm Lik(z)= k , mX=1 m and let −x ∞ n Lik(1 − e ) (k) x −x = Bn . 1 − e nX=0 n! (k) The numbers Bn are the poly-Bernoulli numbers of order k. Kaneko gave a couple of nice formulae for the poly-Bernoulli numbers of negative order, of which one is relevant here. Theorem 3.1 (Kaneko)

min(n,m) (−m) 2 Bn = (j!) S(n +1, j + 1)S(m +1, j + 1). jX=0 This formula has the (entirely non-obvious) corollary that these numbers (−m) (−n) have a symmetry property: Bn = Bm for all non-negative integers n and k. Kaneko’s Theorem together with Theorem 2.1 gives the following result.

Theorem 3.2 The number of acyclic orientations of Kn1,n2 is the poly- (−n2) (−n1) Bernoulli number Bn1 = Bn2 .

7 3.2 Lonesum matrices Another combinatorial interpretation was given by Chad Brewbaker [1] in 2008. A zero-one matrix is a lonesum matrix if it is uniquely determined by its row and column sums. Clearly a lonesum matrix cannot contain either 1 0 0 1 or as a submatrix (in not necessarily consecutive rows or  0 1   1 0  columns). (Since if one such submatrix occurred it could be flipped into the other without changing the row and column sums.) Ryser [4] showed that, conversely, a matrix containing neither of these is a lonesum matrix. Brewbaker showed that the number of n1 × n2 lonesum matrices is given by (−n2) the poly-Bernoulli number Bn1 . We give the simple argument why this number is equal to the number of acyclic orientations of Kn1,n2 . In one direction, number the vertices in the bipartite blocks from 1 to n1 (in A) and from 1 to n2 (in B). Now given an orientation of the graph, we can describe it by a matrix whose (i, j) entry is 1 if the edge from vertex i of A to vertex j of B goes in the direction from A to B, and 0 otherwise. The two forbidden submatrices for lonesum matrices correspond to directed 4-cycles; so any acyclic orientation gives us a lonesum matrix. Conversely, if an orientation of a complete bipartite graph contains no directed 4-cycles, then it contains no directed cycles at all. For suppose that there are no directed 4-cycles, but there is a directed cycle (a1, b1, a2, b2,...,ak, bk, a1). ′ ′ Then the edge between a1 and b must be directed from a1 to b , since other- wise there would be a 4-cycle (a1, b1, a2, b2, a1). But then we have a shorter directed cycle (a1, b2, a3,...,bk, a1). Continuing this shortening process, we would eventually arrive at a directed 4-cycle, a contradiction. (This simply says that the cycle space of the complete bipartite graph is generated by 4-cycles.)

4 Maximizing the number of acyclic orienta- tions

We conjecture that the graphs Kn1,n2 , for |n1 −n2| ≤ 1, maximise the number of acyclic orientations over the class of all graphs with the same numbers of vertices and of edges as these graphs. We further conjecture that for biparite blocks of equal size, the other graphs treated in this paper (that is, Kn,n +e and Kn,n −e) also maximise the

8 number of acyclic orientations for graphs with the same number of vertices and of edges. The evidence and partial results on this are presented in a companion paper [2]. Related to this conjecture, we observed that the ratio

a(Kn,n) − a(Kn,n − e)

a(Kn,n + e) − a(Kn,n) is about 2 within the range of computation; its values for n = 10, 100, 1000 are respectively 1.923534, 1.992995, 1.999306 respectively. The convergence is quite slow; the computed values appear to be 2 − O(n−1). We also observed that, within the range of computation, Kn,n−e has more acyclic orientations than Kn+1,n−1 (these graphs have the same numbers of vertices and edges). For n = 10, 100, 1000, the ratio

a(Kn,n) − a(Kn+1,n−1) a(Kn,n) − a(Kn,n − e) is 1.367903, 1.596801, 1.626101 respectively. It is not so clear how these ratios behave. These considerations will be further discussed in [2].

5 Acknowledgements

Robert Schumacher was funded by EPSRC Grant EP/P504872/1.

References

[1] Chad Brewbaker, A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues, Integers 8 (2008), #A02 (9pp.)

[2] P. J. Cameron, C. A. Glass and R. U. Schumacher, in preparation.

[3] Masanobu Kaneko, Poly-Bernoulli numbers, Journal de Th´eorie des Nombres de Bordeaux 9 (1997), 221–228.

[4] Herbert J. Ryser, Combinatorial properties of matrices of zeros and ones, Canadian J. Math. 9 (1957), 371–377.

9 [5] Richard Stanley, Acyclic orientations of graphs, Discrete Math. 5 (1973), 171–178.

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